MinT - Best Known (81−55, 81, s)-Nets in Base 16

Best Known (81−55, 81, s)-Nets in Base 16

(81−55, 81, 65)-Net over F₁₆ — Constructive and digital

Digital (26, 81, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 81, 65)-net over F₁₆, using
- net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
    - the Hermitian function field over F₁₆ [i]

(81−55, 81, 104)-Net in Base 16 — Constructive

(26, 81, 104)-net in base 16, using

4 times m-reduction [i] based on (26, 85, 104)-net in base 16, using
- base change [i] based on digital (9, 68, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(81−55, 81, 150)-Net over F₁₆ — Digital

Digital (26, 81, 150)-net over F₁₆, using

net from sequence [i] based on digital (26, 149)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 26 and N(F) ≥ 150, using
  - a curve from the manYPoints database [i]

(81−55, 81, 2677)-Net in Base 16 — Upper bound on s

There is no (26, 81, 2678)-net in base 16, because

1 times m-reduction [i] would yield (26, 80, 2678)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 152298 936050 211281 394787 274541 114481 023733 600080 810023 920467 781836 036296 656645 466697 695088 797816 > 16⁸⁰ [i]